Concave univalent functions and Dirichlet finite integral

Abstract

The article deals with the class ${\mathcal{F}}_{\alpha }$ consisting of non-vanishing functions $f$ that are analytic and univalent in $\ID$ such that the complement $\IC\backslash f(\ID) $ is a convex set, $f(1)=\infty ,$ $f(0)=1$ and the angle at $\infty$ is less than or equal to $\alpha \pi ,$ for some $\alpha \in (1,2]$. Related to this class is the class $CO(\alpha )$ of concave univalent mappings in $\ID$, but this differs from ${\mathcal{F}}_{\alpha }$ with the standard normalization $f(0)=0=f^{\prime }(0)=1.$ A number of properties of these classes are discussed which includes an easy proof of the coefficient conjecture for $CO(2)$ settled by Avkhadiev et al. \cite{Avk-Wir-04}. Moreover, another interesting result connected with the Yamashita conjecture on Dirichlet finite integral for $CO(\alpha )$ is also presented.

15 pages; A version of it will appear in the journal "Mathematische Nachrichten"